Discrete Mathematics Course Overview

Discrete Mathematics

• Discrete mathematics is the study of mathematical structures that are countable or otherwise distinct and separable.
• It deals with objects that can consider only distinct, separated values.
• Examples: number of students, houses, integer numbers, etc.

Importance of Discrete Mathematics

• Computer is built on discrete structures.
• Foundation of logic building and problem solving.
• Understanding algorithms and their complexities.
• Software Development.

Propositional Logic

• A proposition is a statement that is either true or false, but not both.
• A proposition is a declarative sentence that has the following properties:
  • It can be either true or false.
  • It cannot be neither.
  • It cannot be both.
• Examples:
  • "The sky is blue" is a proposition because it can be clearly classified as true or false.
  • "4 + 4 = 5" is a proposition because it can be identified as false.

Propositions and Variables

• A proposition can be represented by a variable or symbol.
• Example: P = "Earth is a Planet" and Q = "Earth is a food".
• A proposition can be combined or modified with logical connectives/linkers.

Logical Connectives

• Linkers:
  • And (∧)
  • Or (∨)
  • If (→)
  • If and only if (⇔)
  • Negation / NOT (¬)
• Negation (¬):
  • True if the actual proposition is false.
  • Example: ¬p = "Sakib is not a batsman".

Conjunction and Disjunction

• Conjunction (AND):
  • True if both propositions are true.
  • Example: p ∧ q = "Tamim Iqbal is a batsman and Mustafizur Rahman is a bowler".
• Disjunction (OR):
  • True if any proposition is true.
  • Example: p ∨ q = "Tamim Iqbal is a batsman or Mustafizur Rahman is a bowler".

Exclusive Disjunction (XOR)

• True if exactly one proposition is true.
• Example: p ⊕ q = "Quinton de Kock will play in today's match or Heinrich Klassen will play in today's match".

Truth Tables

• A truth table is used to observe the behavior of a compound proposition at a glance.
• It shows the different possible truth values of the simple propositions and the truth value of the compound one.
• Example: (p ∨ q) ∧ ¬r.

Solving Compound Propositions

• Divide the compound proposition into smaller sections.
• Use the truth table to find the truth value of each section.
• Combine the results to find the final truth value of the compound proposition.

This course covers the fundamentals of discrete mathematics, including logical statements, sets, functions, counting, and graphs. Learn about the study of countable and distinct mathematical structures.